On March 18th in http://cp4space.wordpress.com/2013/03/18/generalising-erdos-conjecture/ Adam P. Goucher writes:
If you can find a combinatorial line of 10 primes, I'll be impressed.
39402*9*7*3 where * takes the values 0 through 9 (same value for each *), i.e. 39402090703, 39402191713, ..., 39402999793, are all primes. The increment had to be divisible by 210, since otherwise at least one of the 10 numbers would have been divisible by 2, 3, 5, or 7. And it had to be all 1s and 0s when expressed in base 10 for the 10 numbers to constitute a combinatorial line by your definition. 101010 was the smallest number which met those criteria. And the above is the smallest set of primes with that increment. That doesn't necessarily mean it's the smallest such set of primes, as there are several other allowable increments less than those primes. Fortuitously, 101010 is also divisible by 13. Otherwise chances would be 10/13 that one of any set of 10 candidate prime numbers is divisible by 13. As it is, chances are 10/11 that one of any set of 10 candidate prime numbers is divisible by 11. I did confirm that no smaller sequence of 10 primes exist with an increment of of 1111110, the smallest increment that's divisible by 2,3,5,7, and 11. (It too is also divisible by 13. Unsurprisingly, given that it's simply 11 times the previous increment.) That makes me wonder, is it always possible to find a number which is all 1s and 0s in base 10 that is divisible by whatever numbers you choose? It's not possible to pick and choose which numbers you want it to be divisible by, given that no number in that form can be divisible by 2 but not by 5 or vice versa. But other than that, is it always possible to pick and choose which primes it is and isn't divisible by? For instance to find a number in that form which is divisible by only the primes (other than 2 and 5, if applicable) that divide your home phone number? That reminds me, some years ago I came up with a proof that you can always find a power of 2 that begins with any desired digit string. In fact, you an always find any number of them. Powers of 2 that begin with your home phone number, powers of 2 that begin, not just with a few billion 1s and 0s, but that begin with the specific pattern of 1s and 0s that comprise your favorite movie DVD. I'm also reminded that on my wall I have a newspaper ad that says, "10101, TENTENONE, The Prime Number." It's an ad for a luxury condo with that street address. Of course 10101 isn't a prime number in base 10 or in base 2. I soon proved it wasn't a prime in any base whatsoever (if the base is an integer and is greater than 1). Getting back to combinatorial lines, the idea seems to be to regard successive digits of an N-digit number as coordinates on successive axes in an N-dimensional space. If several such numbers fall on a line in that space, they constitute a combinatorial line. However, the definition implies that (when the numbers are put in order) the digits in each place must always remain the same or increase in sync. Why can't digits decrease? Why isn't 14 23 32 41 a valid combinatorial line? Why can't some digits increase faster than others? Why isn't 11 23 35 47 a valid combinatorial line?